Modified Logarithmic Sobolev Inequalities and Transportation Cost Inequalities in Rn

In this paper, the modified logarithmic Sobolev inequalities and transportation cost inequalities for measures with density e(-V) in R-n are established. It is proved by using Prekopa-Leindler inequalities following the idea of Bobkov-Ledoux, but a different type of condition is used which recovers Bakry-Emery criterion. As an application, we establish the modified logarithmic Sobolev and transportation cost inequalities for probability measures e(-vertical bar x vertical bar p) dx/Z(p) with p > 1 in R-n, and give out explicit estimates for their constants.